Visualizing fully 3D correlated patterns by fractal geometry
   Anna Carbone   
                                          Visualization has become an essential tool to analyze data generated in diverse fields. Experiment and numerical simulations performed on systems belonging to social sciences, climate modeling, biological tissue are some examples of large amounts of data that exhibit complexity features (self similarity over a wide range of spatial and temporal scales). To the scope of quantify and understand features like formation of patterns, clustered structures and provide a realistic representation of such structure, efficient and accurate analytical tools are required.

Compact fractal disordered fully 3D media are generated and graphically rendered. The obtained structures are "invasive stochastic fractals", with the Hurst exponent varying as a continuous parameter, as opposed to "lacunar deterministic fractals", such as the Menger sponge.  
                                          The Hurst exponent of the generated structure is estimated by using the DMA algorithm. The fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself.

Consequently, in this framework, the Hurst exponent is an estimator of compactness rather than of roughness.

Applications are envisaged for quantifying complex systems characterized by self-similar heterogeneity across space. In particular, we put forward a three-dimensional fractal for snow. The local microstructure of snow is described by a three-dimensional fractional Brownian field and a few relevant physical parameters are quantified in terms of the Hurst exponent.

[1] Algorithm to estimate the Hurst exponent of high-dimensional fractals , A. Carbone, Phys. Rev. E 76, 056703 (2007)

[2] Fractal Heterogeneous Media C. Turk, A. Carbone, B.M. Chiaia, Phys. Rev. E 81, 026706 (2010) [Selected for the "Virtual J. of Nanoscale Science & Technology (http://www.vjnano.org/)" 21, Issue 9 (2010)]

[3] Snow Metamorphism: a Fractal Approach A. Carbone, B.M. Chiaia, B. Frigo, C. Turk, Phys. Rev. E 82, 036103 (2010)